[i]20 problems for 20 minutes.[/i] [b]p1.[/b] Determine how many digits the number $10^{10}$ has. [b]p2.[/b] Let $ABC$ be a triangle with $\angle ABC = 60^o$ and $\angle BCA = 70^o$. Compute $\angle CAB$ in degrees. [b]p3.[/b] Given that $x : y = 2012 : 2$ and $y : z = 1 : 2013$, compute $x : z$. Express your answer as a common fraction. [b]p4.[/b] Determine the smallest perfect square greater than $2400$. [b]p5.[/b] At $12:34$ and $12:43$, the time contains four consecutive digits. Find the next time after 12:43 that the time contains four consecutive digits on a 24-hour digital clock. [b]p6.[/b] Given that $ \sqrt{3^a \cdot 9^a \cdot 3^a} = 81^2$, compute $a$. [b]p7.[/b] Find the number of positive integers less than $8888$ that have a tens digit of $4$ and a units digit of $2$. [b]p8.[/b] Find the sum of the distinct prime divisors of $1 + 2012 + 2013 + 2011 \cdot 2013$. [b]p9.[/b] Albert wants to make $2\times 3$ wallet sized prints for his grandmother. Find the maximum possible number of prints Albert can make using one $4 \times 7$ sheet of paper. [b]p10.[/b] Let $ABC$ be an equilateral triangle, and let $D$ be a point inside $ABC$. Let $E$ be a point such that $ADE$ is an equilateral triangle and suppose that segments $DE$ and $AB$ intersect at point $F$. Given that $\angle CAD = 15^o$, compute $\angle DFB$ in degrees. [b]p11.[/b] A palindrome is a number that reads the same forwards and backwards; for example, $1221$ is a palindrome. An almost-palindrome is a number that is not a palindrome but whose first and last digits are equal; for example, $1231$ and $1311$ are an almost-palindromes, but $1221$ is not. Compute the number of $4$-digit almost-palindromes. [b]p12.[/b] Determine the smallest positive integer $n$ such that the sum of the digits of $11^n$ is not $2^n$. [b]p13.[/b] Determine the minimum number of breaks needed to divide an $8\times 4$ bar of chocolate into $1\times 1 $pieces. (When a bar is broken into pieces, it is permitted to rotate some of the pieces, stack some of the pieces, and break any set of pieces along a vertical plane simultaneously.) [b]p14.[/b] A particle starts moving on the number line at a time $t = 0$. Its position on the number line, as a function of time, is $$x = (t-2012)^2 -2012(t-2012)-2013.$$ Find the number of positive integer values of $t$ at which time the particle lies in the negative half of the number line (strictly to the left of $0$). [b]p15.[/b] Let $A$ be a vertex of a unit cube and let $B$,$C$, and $D$ be the vertices adjacent to A. The tetrahedron $ABCD$ is cut off the cube. Determine the surface area of the remaining solid. [b]p16.[/b] In equilateral triangle $ABC$, points $P$ and $R$ lie on segment $AB$, points $I$ and $M$ lie on segment $BC$, and points $E$ and $S$ lie on segment $CA$ such that $PRIMES$ is a equiangular hexagon. Given that $AB = 11$, $PS = 2$, $RI = 3$, and $ME = 5$, compute the area of hexagon $PRIMES$. [b]p17.[/b] Find the smallest odd positive integer with an odd number of positive integer factors, an odd number of distinct prime factors, and an odd number of perfect square factors. [b]p18.[/b] Fresh Mann thinks that the expressions $2\sqrt{x^2 -4} $and $2(\sqrt{x^2} -\sqrt4)$ are equivalent to each other, but the two expressions are not equal to each other for most real numbers $x$. Find all real numbers $x$ such that $2\sqrt{x^2 -4} = 2(\sqrt{x^2} -\sqrt4)$. [b]p19.[/b] Let $m$ be the positive integer such that a $3 \times 3$ chessboard can be tiled by at most $m$ pairwise incongruent rectangles with integer side lengths. If rotations and reflections of tilings are considered distinct, suppose that there are $n$ ways to tile the chessboard with $m$ pairwise incongruent rectangles with integer side lengths. Find the product $mn$. [b]p20.[/b] Let $ABC$ be a triangle with $AB = 4$, $BC = 5$, and $CA = 6$. A triangle $XY Z$ is said to be friendly if it intersects triangle $ABC$ and it is a translation of triangle $ABC$. Let $S$ be the set of points in the plane that are inside some friendly triangle. Compute the ratio of the area of $S$ to the area of triangle $ABC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Chords $AB$ and $CD$ of a circle intersect inside the circle at $P$ such that $AP = 8, P B = 6, P D = 3$ and $\angle AP C = 60^{\circ}$. Find the area of $\bigtriangleup AP C$.

Javiera and Claudio play on a board consisting of a row with $2019$ cells. Claudio starts by placing a token anywhere on the board. Next Javiera says a natural number $k$, $1 \le k \le n$ and Claudio must move the token to the right or to the left at your choice $k$ squares and so on. Javiera wins if she manages to remove the piece that Claudio moves from the board. Determine the smallest $n$ such that Javiera always wins after a finite number of moves.

Let the $f:\mathbb{N}\rightarrow\mathbb{N}$ be the function such that (i) For all positive integers $n,$ $f(n+f(n))=f(n)$ (ii) $f(n_o)=1$ for some $n_0$ Prove that $f(n)\equiv 1.$

Compute the number of positive integers $n\le1000$ such that $\text{lcm}(n,9)$ is a perfect square. (Recall that $\text{lcm}$ denotes the least common multiple.)

Let $f(x)$ and $g(x)$ be degree $n$ polynomials, and $x_0,x_1,\ldots,x_n$ be real numbers such that $$f(x_0)=g(x_0),f'(x_1)=g'(x_1),f''(x_2)=g''(x_2),\ldots,f^{(n)}(x_n)=g^{(n)}(x_n).$$Prove that $f(x)=g(x)$ for all $x$.

Let $n$ be a positive integer greater than $1$. Evaluate the following summation: \[ \sum_{k=0}^{n-1} \frac{1}{1 + 8 \sin^2 \left( \frac{k \pi}{n} \right)}. \]

A $3$ by $3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant.

Let $m, n$ be positive integers. Let $S(n,m)$ be the number of sequences of length $n$ and consisting of $0$ and $1$ in which there exists a $0$ in any consecutive $m$ digits. Prove that \[S(2015n,n).S(2015m,m)\ge S(2015n,m).S(2015m,n)\]

In a sports tournament involving $N$ teams, each team plays every other team exactly one. At the end of every match, the winning team gets $1$ point and losing team gets $0$ points. At the end of the tournament, the total points received by the individual teams are arranged in decreasing order as follows: \[x_1 \ge x_2 \ge \cdots \ge x_N . \] Prove that for any $1\le k \le N$, \[\frac{N - k}{2} \le x_k \le N - \frac{k+1}{2}\]

The cricles $\Gamma_1$ and $\Gamma_2$ intersect in $ M $ and $N.$ A line that goes through $ M $ intersects the cricles $\Gamma_1$ and $\Gamma_2$ in $ A$ and $B$, such that $M\in(AB)$. The bisector of angle $ AMN $ intersects the circle $\Gamma_1$ in $D,$ and the bisector of angle $BMN$ intersects the circle $\Gamma_2$ in $C.$ Prove that the circle with diameter $CD$ splits the segment $AB$ in half.

Let \(m\) be a positive integer. Find, in terms of \(m\), all polynomials \(P(x)\) with integer coefficients such that for every integer \(n\), there exists an integer \(k\) such that \(P(k)=n^m\). [i]Proposed by Raymond Feng[/i]

[b]p1.[/b] Trung took five tests this semester. For his first three tests, his average was $60$, and for the fourth test he earned a $50$. What must he have earned on his fifth test if his final average for all five tests was exactly $60$? [b]p2.[/b] Find the number of pairs of integers $(a, b)$ such that $20a + 16b = 2016 - ab$. [b]p3.[/b] Let $f : N \to N$ be a strictly increasing function with $f(1) = 2016$ and $f(2t) = f(t) + t$ for all $t \in N$. Find $f(2016)$. [b]p4.[/b] Circles of radius $7$, $7$, $18$, and $r$ are mutually externally tangent, where $r = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p5.[/b] A point is chosen at random from within the circumcircle of a triangle with angles $45^o$, $75^o$, $60^o$. What is the probability that the point is closer to the vertex with an angle of $45^o$ than either of the two other vertices? [b]p6.[/b] Find the largest positive integer $a$ less than $100$ such that for some positive integer $b$, $a - b$ is a prime number and $ab$ is a perfect square. [b]p7.[/b] There is a set of $6$ parallel lines and another set of six parallel lines, where these two sets of lines are not parallel with each other. If Blythe adds $6$ more lines, not necessarily parallel with each other, find the maximum number of triangles that could be made. [b]p8.[/b] Triangle $ABC$ has sides $AB = 5$, $AC = 4$, and $BC = 3$. Let $O$ be any arbitrary point inside $ABC$, and $D \in BC$, $E \in AC$, $F \in AB$, such that $OD \perp BC$, $OE \perp AC$, $OF \perp AB$. Find the minimum value of $OD^2 + OE^2 + OF^2$. [b]p9.[/b] Find the root with the largest real part to $x^4-3x^3+3x+1 = 0$ over the complex numbers. [b]p10.[/b] Tony has a board with $2$ rows and $4$ columns. Tony will use $8$ numbers from $1$ to $8$ to fill in this board, each number in exactly one entry. Let array $(a_1,..., a_4)$ be the first row of the board and array $(b_1,..., b_4)$ be the second row of the board. Let $F =\sum^{4}_{i=1}|a_i - b_i|$, calculate the average value of $F$ across all possible ways to fill in. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$. Prove that the triangle is isosceles.

A container is filled with a total of $51$ red and white balls and has at least $1$ red ball and $1$ white ball. The probability of picking up $3$ red balls and $1$ white ball, without replacement, is equivalent to the probability of picking up $1$ red ball and $2$ white balls, without replacement. Compute the original number of red balls in the container.

Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.

A positive natural number $n$ is said to be [i]comico[/i] if its prime factorization is $n = p_1p_2...p_k$, with $k\ge 3$, and also the primes $p_1,..., p_k$ they fulfill that $p_1 + p_2 = c^2_1$ $p_1 + p_2 + p_3 = c^2_2$ $...$ $p_1 + p_2 + ...+ p_n = c^2_{n-1}$ where $c_1, c_2, ..., c_{n-1}$ are positive integers where $c_1$ is not divisible by $7$. Find all comico numbers less than $10,000$.

Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedron is exactly edge $AB$.

Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$, \[a^n + a^{n-1} +
\cdots
+ a^1 + 1 \mid a^{n!} + a^{(n-1)!} +
\cdots

+ a^{1!} + 1.\] [i]Proposed by Milos Milosavljevic[/i]

Given that $x$ and $y$ are positive real numbers such that $\frac{5}{x}=\frac{y}{13}=\frac{x}{y}$, find the value of $x^3 + y^3$. Proposed by Ephram Chun

Let $ABC$ be a triangle, and let $D$ and $D_1$ be points on segment $BC$ such that $BD = CD_1$. Construct point $E$ such that $EC\perp BC$ and $ED\perp AC$. Similarly, construct point $F$ such that $FB\perp BC$ and $FD\perp AB$. Prove that $EF\perp AD_1$.

$n$ numbers are written down along the circumference. Their sum equals to zero, and one of them equals $1$. a) Prove that there are two neighbours with their difference not less than $n/4$. b) Prove that there is a number that differs from the arithmetic mean of its two neighbours not less than on $8/(n^2)$. c) Try to improve the previous estimation, i.e what number can be used instead of $8$? d) Prove that for $n=30$ there is a number that differs from the arithmetic mean of its two neighbours not less than on $2/113$, give an example of such $30$ numbers along the circumference, that not a single number differs from the arithmetic mean of its two neighbours more than on $2/113$.

Find the number of all real functions $f$ which map the sum of $n$ elements into the sum of their images, such that $f^{n-1}$ is a constant function and $f^{n-2}$ is not. Here $f^0(x) = x$ and $f^k = f \circ f^{k-1}$ for $k \ge 1$.

Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling, Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning? $\textbf{(A)}\ 20 \qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 35\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 45$

Prove that in each year , the $13^{th}$ day of some month occurs on a Friday .